,
and 
and 
.
and 
be the intersection of 
,
and 
are perpendicular.
that verify the following equation for all 
:
.
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
be the circle with diameter 
.
be the intersection of 
and 
.
be the intersection of 
and 
be the lines through 
respectively. Let 
and 
.
and 
.
be the line through 
and let 
be the reflection of 
is tangent to 
.
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
with circumcenter 
is constant.
that satisfy the equation 
.
intersect 
Circles 
are internally tangent to 
and 
respectively, and both are tangent to 
Similarly, circles 
and 
are externally tangent to 
respectively.
the segment 
has a length of 
and 
find the length of segment 
and 
Prove that
.
Prove that
and 
Prove that
Y by giveandtake
In an acute-angled triangle 
, points 
are the feet of altitudes drawn from the vertices 
, respectively, and 
is the point of intersection of altitudes. Let point 
be the midpoint of the 
be the intersection point of the segments 
and 
, and 
be the intersection point of the straight line 
and side 
. Prove that the line 
is perpendicular to the side 
.
, points 
are the feet of altitudes drawn from the vertices 
, respectively, and 
is the point of intersection of altitudes. Let point 
be the midpoint of the 
be the intersection point of the segments 
and 
, and 
be the intersection point of the straight line 
and side 
. Prove that the line 
is perpendicular to the side 
.This post has been edited 3 times. Last edited by parmenides51, Jun 21, 2022, 2:58 AM
and 
then 
and then 
and 
is cyclic, so 
